Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : Time period of a simple pendulum is longer at the top of a mountain than that at the base of the mountain. Reason (R): Time period of a simple pendulum decreases with increasing value of acceleration due to gravity and vice-versa. In the light of the above statements, choose the most appropriate answer from the options given below :

We need to evaluate the Assertion and Reason about a simple pendulum on a mountain.

Assertion (A): "Time period of a simple pendulum is longer at the top of a mountain than that at the base of the mountain."

Analysis: The time period of a simple pendulum is given by:

$$T = 2\pi\sqrt{\frac{l}{g}}$$

where $$l$$ is the length and $$g$$ is the acceleration due to gravity. At the top of a mountain, the distance from the Earth's center increases, so $$g$$ decreases (since $$g = GM/R^2$$ and the effective $$R$$ increases). With a smaller $$g$$, the time period $$T$$ increases (longer period). Assertion (A) is TRUE.

Reason (R): "Time period of a simple pendulum decreases with increasing value of acceleration due to gravity and vice-versa."

Analysis: From the formula $$T = 2\pi\sqrt{l/g}$$, we see that $$T \propto 1/\sqrt{g}$$. As $$g$$ increases, $$T$$ decreases, and as $$g$$ decreases, $$T$$ increases. Reason (R) is TRUE.

Relationship: The Reason directly explains the Assertion. The pendulum has a longer period on the mountain because $$g$$ is smaller there, and as R states, a smaller $$g$$ leads to a longer time period. R is the correct explanation of A.

The correct answer is Option 1: Both (A) and (R) are true and (R) is the correct explanation of (A).